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(a)
A
C
(b)
A
B
C
D
B
D
AB
C
+
ĺ
D
AB
C
+
ĺ
D
ĺ
ĺ
ĺ
A
B
||
||
+
B
ĺĺ
+
ĺĺ
+
+
ĺ
C
+
+
C
AB
C
+
ĺ
D
D
+
D
+
ĺ
ĺ
B
ĺĺ
+
C
+
+
+
D
+
(c)
A
B
C
D
Fig. 5.
The behavioural profile of model (c) corresponds to the intersection of the
profiles of models (a) and (b)
sets of traces of models (a) and (b). Referring to our example in Fig. 1, the set of
shared complete traces is empty for both models. The evident behavioural com-
monalities of both models are not revealed by a trace-based assessment. The in-
tersection of behavioural profiles allows to address such scenarios.
Union.
The union operation for two behavioural profiles yields a third be-
havioural profile that combines the weakest constraints of the two profiles given
as input parameters for all pairs of activities. We trace back the definition of the
union operation to the complement and intersection operations using De Mor-
gan's rule. The union
B
3
of two behavioural profiles
B
1
and
B
2
over a set of
activities
A
, denoted by
B
3
=
B
1
∪B
2
is defined as
B
3
=
B
1
∩ B
2
.
4 Managing Process Variability
In this section, we discuss how the set algebra for behavioural profiles is applied
to address the questions raised in Section 2.1 regarding the management of
decoupled process variants. Let
P
1
,...,P
n
be a set of models that were found to
capture different variantsofabusinessprocesswith
A
being the shared activities.
C1: Shared behaviour: Greatest Common Divisor.
The shared behaviour may
be referred to as the Greatest Common Divisor (GCD), cf., [14]. Given a set of
process variants, the GCD is characterised by a behavioural profile
B
GCD
over
A
that is derived by computing the intersection of all profiles
B
P
n
. Hence,
the GCD integrates the constraints shared by all variants. The profile
B
P
1
,...,
B
GCD
may be checked for emptiness. If it is empty, all variants impose contradicting
constraints for the shared activities. A model representing the GCD can be syn-
thesized from the profile
B
GCD
following the approach introduced in [26]. Note
that the synthesises imposes certain consistency requirements on the behavioural
profile. Further, the behavioural commonality between a single variant
P
i
and
the GCD is quantified as the relative share of activity pairs that have equal rela-
tions in
B
P
i
. This measure quantifies how much additional behaviour
the variant allows for, relative to the behaviour shared with all variants.
B
GCD
and
